Subgroups — #147100

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#368	C₂	1	1	yes	maximal, characteristic
2	C₂²×#367	C₂	1	1	yes	maximal, characteristic
3	C₂×#2108	C₂	1	1	yes	maximal, characteristic
4	C₂×#1387	C₂	1	2	yes	maximal
5	#10870	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	1	yes	characteristic
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂²×#107	C₂²	1	2	yes	—
10	C₂×#405	C₂²	1	1	yes	characteristic
11	C₂×#368	—	2	2	no	—
12	C₂×#368	—	2	2	no	—
13	C₂×#368	C₄	1	4	yes	complemented
14	C₂×#367	—	2	1	no	—
15	C₂×#367	—	2	1	no	—
16	C₂×#367	—	2	2	no	—
17	C₂×#367	C₄	1	4	yes	complemented
18	#2108	—	2	2	no	—
19	#2108	C₄	1	4	yes	complemented
20	C₂×#346	—	2	1	no	—
21	C₂×C₄×C₁₆	—	2	1	no	—
22	C₂×#343	—	2	2	no	—
23	#1736	—	2	2	no	—
24	#1387	—	2	4	no	—
25	C₂²×M₁₆	—	2	2	no	—
26	C₂²×M₁₆	D₄	1	1	yes	characteristic
27	C₂³×C₈	C₂³	1	1	yes	characteristic
28	C₂×#117	D₄	1	1	yes	characteristic
29	C₂²×C₄⋊C₄	—	2	2	no	—
30	C₂×M₃₂	—	2	1	no	—
31	C₂×M₃₂	—	2	1	no	—
32	C₂×M₃₂	—	2	1	no	—
33	C₂×M₃₂	C₂×C₄	1	2	yes	—
34	C₂×M₃₂	D₄	1	2	yes	—
35	C₂×M₃₂	Q₈	1	2	yes	—
36	C₂²×C₁₆	—	2	1	no	—
37	C₂²×C₁₆	C₂×C₄	1	2	yes	—
38	C₂²×C₁₆	C₂×C₄	1	2	yes	—
39	C₂²×C₁₆	C₂×C₄	1	2	yes	—
40	C₂²×C₁₆	D₄	1	1	yes	characteristic
41	C₂²×C₁₆	D₄	1	1	yes	characteristic
42	C₂²×C₁₆	D₄	1	2	yes	—
43	C₂²×C₁₆	D₄	1	2	yes	—
44	C₂²×C₁₆	Q₈	1	2	yes	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	—	2	2	no	—
47	C₂×#108	—	2	2	no	—
48	C₂×#108	—	2	2	no	—
49	C₂×#108	—	2	2	no	—
50	C₂×#108	C₂×C₄	1	4	yes	—
51	C₂×#107	—	2	2	no	—
52	C₂×#107	—	2	2	no	—
53	C₂×#107	—	2	2	no	—
54	C₂×#107	—	2	2	no	—
55	C₂×#107	—	2	2	no	—
56	C₂×#107	C₂×C₄	1	4	yes	—
57	#405	—	2	1	no	—
58	#405	—	2	1	no	—
59	#405	C₂×C₄	1	2	yes	—
60	#405	Q₈	1	2	yes	—
61	C₂×#97	—	2	1	no	—
62	C₂×C₄×C₈	—	2	1	no	—
63	#368	—	2	8	no	—
64	#367	—	2	4	no	—
65	#367	—	2	4	no	—
66	#346	—	2	4	no	—
67	C₄×C₁₆	—	2	4	no	—
68	#343	—	4	4	no	—
69	#340	—	4	4	no	—
70	C₂³×C₄	—	4	2	no	—
71	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×M₁₆	—	2	1	no	—
73	C₂×M₁₆	—	2	1	no	—
74	C₂×M₁₆	—	2	1	no	—
75	C₂×M₁₆	—	2	4	no	—
76	C₂×M₁₆	—	4	2	no	—
77	C₂×M₁₆	—	4	2	no	—
78	C₂×M₁₆	—	4	2	no	—
79	C₂×M₁₆	—	4	2	no	—
80	C₂×M₁₆	—	4	2	no	—
81	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
82	C₂×M₁₆	D₈	1	2	yes	—
83	C₂×M₁₆	Q₁₆	1	2	yes	—
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	—	4	2	no	—
86	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₈	#41	1	1	yes	characteristic
97	C₂²×C₈	#41	1	1	yes	characteristic
98	C₂²×C₈	#41	1	1	yes	characteristic
99	C₂²×C₈	#41	1	1	yes	characteristic
100	#117	—	2	1	no	—
101	#117	—	2	1	no	—
102	#117	C₄⋊C₄	1	2	yes	—
103	#117	Q₁₆	1	2	yes	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	—	4	2	no	—
106	C₂×C₄⋊C₄	—	4	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄⋊C₄	—	4	2	no	—
109	C₂×C₄⋊C₄	—	4	2	no	—
110	C₂×C₄⋊C₄	D₈	1	2	yes	—
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	—	2	1	no	—
113	M₃₂	—	2	2	no	—
114	M₃₂	—	2	2	no	—
115	M₃₂	—	2	2	no	—
116	M₃₂	C₄⋊C₄	1	4	yes	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	C₄²	1	4	yes	—
121	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
122	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
123	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
124	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
125	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
126	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
127	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
128	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
129	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
130	#108	—	2	4	no	—
131	#108	—	2	4	no	—
132	#108	—	4	2	no	—
133	#108	—	4	2	no	—
134	#107	—	2	4	no	—
135	#107	—	2	4	no	—
136	#107	—	4	2	no	—
137	#107	—	4	2	no	—
138	#97	—	2	2	no	—
139	#97	—	2	2	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₂⁴	#135	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	—	8	2	no	—
146	C₂²×C₄	—	8	2	no	—
147	C₂²×C₄	—	16	4	no	—
148	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
150	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
151	C₂²×C₄	#121	1	1	yes	characteristic
152	C₂²×C₄	#121	1	1	yes	characteristic
153	C₂²×C₄	#127	1	1	yes	characteristic
154	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
157	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
158	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
159	C₂²×C₄	#135	1	1	yes	characteristic
160	C₂²×C₄	#135	1	1	yes	characteristic
161	C₂²×C₄	#135	1	1	yes	characteristic
162	M₁₆	—	2	2	no	—
163	M₁₆	—	2	2	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	—	4	4	no	—
166	M₁₆	—	4	4	no	—
167	M₁₆	—	8	2	no	—
168	M₁₆	—	8	2	no	—
169	M₁₆	#107	1	4	yes	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	8	2	no	—
175	C₂×C₈	—	8	2	no	—
176	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₈	#117	1	1	yes	characteristic
185	C₂×C₈	#117	1	1	yes	characteristic
186	C₂×C₈	C₄×D₄	1	1	yes	characteristic
187	C₂×C₈	C₄×D₄	1	1	yes	characteristic
188	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
189	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
190	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
191	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
192	C₂×C₈	#121	1	1	yes	characteristic
193	C₂×C₈	#121	1	1	yes	characteristic
194	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
197	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
198	C₂×C₈	#124	1	1	yes	characteristic
199	C₂×C₈	#124	1	1	yes	characteristic
200	C₂×C₈	#125	1	1	yes	characteristic
201	C₂×C₈	#125	1	1	yes	characteristic
202	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
203	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	8	2	no	—
208	C₄⋊C₄	—	8	2	no	—
209	C₄⋊C₄	#107	1	4	yes	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₁₆	—	2	4	no	—
215	C₁₆	#95	1	4	yes	—
216	C₁₆	#95	1	4	yes	—
217	C₂³	—	2	4	no	—
218	C₂³	#427	1	1	yes	characteristic
219	C₂³	C₄×D₈	1	1	yes	characteristic
220	C₂³	C₄×Q₁₆	1	1	yes	characteristic
221	C₂³	#466	1	1	yes	characteristic
222	C₂³	#467	1	1	yes	characteristic
223	C₂³	#495	1	1	yes	characteristic
224	C₂³	#499	1	1	yes	characteristic
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	8	4	no	—
228	C₂×C₄	—	16	4	no	—
229	C₂×C₄	—	16	4	no	—
230	C₂×C₄	—	16	4	no	—
231	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
232	C₂×C₄	#380	1	1	yes	characteristic
233	C₂×C₄	#384	1	1	yes	characteristic
234	C₂×C₄	#389	1	1	yes	characteristic
235	C₂×C₄	#389	1	1	yes	characteristic
236	C₂×C₄	#390	1	1	yes	characteristic
237	C₂×C₄	#390	1	1	yes	characteristic
238	C₂×C₄	C₂×#107	1	1	yes	characteristic
239	C₂×C₄	C₂×#107	1	1	yes	characteristic
240	C₂×C₄	#427	1	1	yes	characteristic
241	C₂×C₄	C₄×D₈	1	1	yes	characteristic
242	C₂×C₄	C₄×D₈	1	1	yes	characteristic
243	C₂×C₄	C₄×D₈	1	1	yes	characteristic
244	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
245	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
246	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
247	C₂×C₄	#466	1	1	yes	characteristic
248	C₂×C₄	#466	1	1	yes	characteristic
249	C₂×C₄	#466	1	1	yes	characteristic
250	C₂×C₄	#467	1	1	yes	characteristic
251	C₂×C₄	#467	1	1	yes	characteristic
252	C₂×C₄	#467	1	1	yes	characteristic
253	C₂×C₄	#493	1	1	yes	characteristic
254	C₂×C₄	#494	1	1	yes	characteristic
255	C₂×C₄	#495	1	1	yes	characteristic
256	C₂×C₄	#499	1	1	yes	characteristic
257	C₂×C₄	#500	1	1	yes	characteristic
258	C₂×C₄	#500	1	1	yes	characteristic
259	C₈	—	2	4	no	—
260	C₈	—	8	4	no	—
261	C₈	C₂×#95	1	1	yes	characteristic
262	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
263	C₈	#380	1	1	yes	characteristic
264	C₈	#382	1	1	yes	characteristic
265	C₈	#389	1	1	yes	characteristic
266	C₈	#389	1	1	yes	characteristic
267	C₈	#391	1	1	yes	characteristic
268	C₈	#391	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	C₄×#107	1	1	yes	characteristic
276	C₂²	#1808	1	1	yes	characteristic
277	C₂²	#1846	1	1	yes	characteristic
278	C₂²	#1942	1	1	yes	characteristic
279	C₂²	#1942	1	1	yes	characteristic
280	C₂²	#1969	1	1	yes	characteristic
281	C₂²	#1970	1	1	yes	characteristic
282	C₂²	#2223	1	4	yes	—
283	C₄	—	4	4	no	—
284	C₄	—	16	4	no	—
285	C₄	C₄⋊#95	1	1	yes	characteristic
286	C₄	C₄×#107	1	1	yes	characteristic
287	C₄	#1808	1	1	yes	characteristic
288	C₄	#1845	1	1	yes	characteristic
289	C₄	#1942	1	1	yes	characteristic
290	C₄	#1942	1	1	yes	characteristic
291	C₄	#1969	1	1	yes	characteristic
292	C₄	#1970	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	#10862	1	1	yes	characteristic
298	C₂	#12843	1	2	yes	—
299	C₂	#13172	1	2	yes	—
300	C₂	#13246	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
13. C₂×#368	C₄	semidirect
17. C₂×#367	C₄	semidirect
19. #2108	C₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#368	2
C₂²×#367	2
C₂×#2108	2
C₂×#1387	2
#10870	2